The equation $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ is a formula used in set theory to find the number of elements in the union of two sets, $A$ and $B$. Let's break down each part of this equation and explain it with the help of a Venn diagram.

Explanation of Terms:

1. $A \cup B$: The union of sets $A$ and $B$ includes all elements that are in $A$, $B$, or both. We want to know the total count of unique elements in this union.

2. $n(A)$: This represents the number of elements in set $A$.

3. $n(B)$: This represents the number of elements in set $B$.

4. $A \cap B$: The intersection of sets $A$ and $B$ includes only those elements that are present in both sets $A$ and $B$.

5. $n(A \cap B)$: This represents the number of elements in the intersection of $A$ and $B$, or the count of elements shared by both sets.

Why We Subtract $n(A \cap B)$:

When we add $n(A)$ and $n(B)$ together, we count each element in $A \cap B$ twice — once as part of $A$ and once as part of $B$. To avoid this double-counting, we subtract $n(A \cap B)$, so each element is counted only once in the total.

Visual Explanation with Venn Diagram

Imagine a Venn diagram with two overlapping circles:

1. Circle $A$ represents set $A$.
2. Circle $B$ represents set $B$.
3. The overlapping area of the two circles represents $A \cap B$ — elements that are in both $A$ and $B$.
4. The area covered by both circles represents $A \cup B$ — all elements in either $A$ or $B$, including their intersection.

Here’s how the formula works visually:

• $n(A)$: Count of all elements within Circle $A$, including those in the overlap.
• $n(B)$: Count of all elements within Circle $B$, including those in the overlap.
• Subtracting $n(A \cap B)$ removes the overlap area, ensuring we don't count those elements twice.

Example

Suppose:

• $n(A) = 5$
• $n(B) = 6$
• $n(A \cap B) = 2$

Using the formula: $n(A \cup B) = 5 + 6 - 2 = 9$

Summary

The formula $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ ensures that elements in the overlap are only counted once in the total count of elements in the union.

Would you like to see further details or have any questions?

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Related Questions

1. How can this formula be extended to three sets?
2. What is the significance of the intersection in probability?
3. How does the inclusion-exclusion principle apply in set theory?
4. How can we use this formula in real-world applications, like survey data?
5. What happens when two sets have no intersection?

Tip

For three or more sets, the inclusion-exclusion principle helps adjust for overlapping elements across multiple sets, accounting for all shared elements without double-counting.